A challenge in packaging is undesired package resonances. The cause of these resonances is a volume that is enclosed by electric conductors to create a cavity. All metal cavities filled with a vacuum will display a resonance at a frequency given by
Cavity Resonant Frequency = fr=c/2a (1)
fr = frequency of the resonance (Hz)
a = width of the cavity (m)
c = speed of light in a vacuum (m/s)
The cavity resonance predicted by (1) is the lowest frequency of resonance and can be a major concern for those developing microwave hybrids or Microwave Integrated Circuits (MICs).
The cross section of a microwave hybrid is illustrated in the figure below. It shows the metal box, dielectric substrate and internal components. As can be seen in figure, the electric field distribution has its peak in the middle and decays to zero and the metallic walls of the metal box. The electric field is such that it will set up a package resonance that will cause RF “suck outs” to occur.
The lowest frequency at which the module will resonant is the most important to know and to manage. This is due to the simple fact that if the lowest resonant mode is avoided or managed properly, the other higher modes in most cases can be ignored. Thankfully, the lowest resonant mode can be calculated or approximated for most electronic models enclosed by metal walls. This can be useful to determine the proper dimensions of the housing to avoid the lowest resonance.
A first order approximation of the resonant frequency can be achieved by ignoring the dielectric substrate and treating the cavity as a waveguide. In this case it is simple to calculate the resonant frequency using (1). Such an approximation will give a resonant frequency prediction which is actually higher than will be measured in the module. However, since calculation is so easily done by hand, it offers a quick assessment of resonances.
A more precise answer is usually needed which requires the dielectric substrate to be taken into account. This is achieved by recognizing the package resonance is due to the propagation of the Longitudinal Section Magnetic (LSM) mode. In particular, it is the LSM11 mode.
A variational method can be applied to solve for the propagation constant of a dielectric filled waveguide which will give us the desired resonant frequency of the cavity. The equations are given in [1, 2] and a Python program was written to solve it. You may download the Python code (in Jupyter format file) here. and screen shots of the code are below.
 R.E. Collin, Field Theory of Guided Waves, 2nd Edition, Piscataway, NJ: IEEE Press, 1991, pp 428-429.
 R.L. Sturdivant, Microwave and Millimeter-Wave Electronic Packaging, Norwood, MA: Artech House, 2013.